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The “control diagram” method
Chapter 8
The “control diagram” method 8.1 A short introduction to the “control diagram” method 8.1.1
A third approach to solving the “Swing Equation”
In addition to the “exact solution of the differential equation” and “Laplace transform” methods covered in Chapter 6, Analytical solutions, and the “numerical solution” methods of solving the “swing equation” covered in Chapter 7, Numerical methods for solving the “swing equation,” of this book, there is a third possible approach available, which may be termed the “control diagram” method. This approach couches the problem of the solution of the “swing equation” in the language of “control engineering” by using, instead of algebraic and differential equations, “block diagrams” or “control diagrams.” For examples of this approach, please see Refs. [3] and [4] listed at the end of this chapter. The problem is the same as for the other methods, namely solving Eq. (7.1), but the way the solution is approached is slightly different from that of the other methods.
8.1.2 The use of the “control diagram” to represent a differential equation The idea of using a “control diagram” to represent a differential equation such as the “swing equation” is, of course, by no means new (please see, for example, Ref. [1], Figures 11.4 and 11.5 on p. 585 for a basic illustration of the “control diagram” representation of the elementary “swing equation” with the addition of a single feedback loop to model the effect of “load damping,” that is, the variation of load with frequency). However, it can be a useful and visually appealing means of better understanding the relationships between the various physical and logical elements in an algebraic-differential equation such as the “swing equation.” For example, it groups the sum of all the sources of generation minus the demand into one element: the input, and the change in frequency as the output. In this way it separates out “cause” and “effect” by representing the system frequency equations as “input” and “output,” and it represents the damping-type terms all together as one feedback loop. Modern Aspects of Power System Frequency Stability and Control. DOI: https://doi.org/10.1016/B978-0-12-816139-5.00008-4 © 2019 Elsevier Inc. All rights reserved.
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Development of the “control diagram” representation
The basic representation of the control diagram given in Ref. [1], Figures 11.4 and 11.5 on p. 585 can be extended to include the extra features contained in our “standard version” of the “swing equation” given in Eqs. (8.1a)(8.1d). Thus in addition to the features contained already in Figures 11.4 and 11.5 on p. 585 of Ref. [1], namely the system power balance including the variation of load with frequency (load damping) and generator “governor response,” we can also include further elements, such as the “general responsive profile”, represented in Eqs. (8.1a)(8.1d) by a0 1 a1 t. However, representing the unit step-change term PTRIP u(t 2 tTRIP) of Eqs. (8.1a)(8.1d) may prove more challenging using the “control system” method.
8.2 Recalling the “swing equation” 8.2.1
The “standard form” of the “swing equation”
Let us quote, for convenience, and for further use in the present chapter, the “standard form” of the “swing equation” that was last given in the previous chapter as [Eq. (7.1)]: ΣPGENðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n 2 k1 Uðf fn Þ 2 PTRIP Uuðt tTRIP Þ df 1 a0 1 a1 t 5 2UHUf0 U dt ð8:1aÞ
8.2.2
A simplifying substitution
We can simplify [Eq. (8.1a)] further for our purposes in this chapter by making the substitution: Δf f fn
ð8:1bÞ
so that Eq. (8.1a) may be rewritten in a more convenient form as 2UHUf0 U
dðΔf Þ 1 k1 UðΔf Þ 5 ΣPGEN ðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n dt
ð8:1cÞ
2 PTRIP Uuðt tTRIP Þ 1 a0 1 a1 t where in addition we have taken advantage of the fact that dðΔf Þ df 5 dt dt which is true since we have dfn/dt 5 0.
ð8:1dÞ
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All that we have done here is to change the dependent variable from f to Δf because this fits in better with the “block diagram” representation we are going to use.
8.2.3
In the next subsection
In the next subsection we are going to start to look at how to formulate the “control system” method, beginning with Eq. (8.1c). This involves first representing the “swing equation” in “state-space format.” This is an intermediate step between the algebraic-differential representation we are starting with and the “control diagram” or “block diagram” representation we are going to finish up with.
8.3 The representation of the “swing equation” in “state-space” format 8.3.1
An introduction to “state-space” format
First, the equation [Eqs. (8.1a)(8.1d)] can be expressed, in a completely equivalent way, in the following well-known “state-space” format, which may be defined as follows: x_ 5 AUx 1 BUu
ð8:2aÞ
y 5 CUx 1 DUu
ð8:2bÞ
and
(see, for example, Ref. [1], p. 704 for a more detailed description of this representation). In Eqs. (8.2a) and (8.2b) the variable u is the “input variable,” the variable y is the “output variable,” and the variable x is the “system variable” or “state variable.” The parameters A, B, C, and D are constants. The “state-space” format has the in-built flexibility for choosing the variables and constants to suit a desired application—a flexibility that we shall be making use of in the next section.
8.3.2 Application of the “state-space” format to the “swing equation” To represent the “swing equation,” in the form represented by Eq. (8.1c), in the “state-space” format of Eqs. (8.2a) and (8.2b), we may select x 5 y 5 Δf 5 f fn
ð8:2cÞ
as the output variable. This represents the amount that the “system frequency” differs from the “nominal frequency.”
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Following Ref. [2], p. 753 and using their notation, our version of the “swing equation” given in Eq. (8.1c) may be written in the form: a0 Uy_ 1 a1 Uy 5 b1 Uu
ð8:3aÞ
so that, in their notation, we have a0 5 2UHUf0
ð8:3bÞ
a1 5 k 1
ð8:3cÞ
b1 5 1
ð8:3dÞ
and we may take
For the input variable u we therefore have u 5 ΣPGEN ðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n 2 PTRIP Uuðt tTRIP Þ 1 a0 1 a1 t ð8:3eÞ Thus we have provided values for the three parameters A, B, and u of Eq. (8.2a) in addition to taking C 5 1 and D 5 0 in Eq. (8.2b) to complete the conversion of the “swing equation” into “state-space” format. In the next section, we shall provide an interpretation of some of the terms in the above set of equations.
8.3.3 An interpretation of some of the terms in Eqs. (8.3a)(8.3e) inclusive 1. A comment on the information distribution of Eqs. (8.3a)(8.3e) We can see from the list of equations earlier that most of the information for the “state-space” format of the “swing equation” is contained in Eqs. (8.3c) and (8.3e). We now therefore proceed to look more closely at these two equations, taking them in turn. 2. The information contained in Eq. (8.3c) In Eq. (8.3c), there is information contained about the dependency of the “system frequency” f on itself since it multiplies y in Eq. (8.3a). It is therefore a kind of “feedback” term. We shall see that this assertion will be confirmed graphically later in our analysis. We recall from previous chapters, for example, in Eq. (7.1a), that the parameter k1 is given by k1 5 ΣPLOAD n Uα 1 ΣPGEN n ðRÞ Uk
ð8:3fÞ
where ΣPLOAD n represents the total MW load at nominal frequency, α is the “load-damping coefficient,” ΣPGEN n (R) represents the total MW generation of responsive synchronous machines at nominal frequency, and k is a parameter essentially equal to the reciprocal of the slope D of
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the synchronous machine governor characteristic [see, for example, Eq. (A.9) of Appendix A for the full definition of this parameter]. 3. The information contained in Eq. (8.3e) In Eq. (8.3e), we have all the information relating to the net input to the system: generation minus demand (or load: demand and load are the same here because we are ignoring I2R losses). We can divide Eq. (8.3e) into the following groups of terms: a. The basic “power balance” term at “nominal frequency” The term “ΣPGEN (NR) 1 ΣPGEN n (R) 2 ΣPLOAD n” in Eq. (8.3e) represents the basic power balance at the “nominal frequency” of the system in the absence of any “trips” and before any “responsive output” has appeared on the system. b. In the absence of any responsive output only If “ΣPGEN (NR) 1 ΣPGEN n (R)” is less than “ΣPLOAD n” then the frequency will settle down to a value lower than the “nominal frequency.” If “ΣPGEN (NR) 1 ΣPGEN n (R)” equals “ΣPLOAD n” then the frequency will settle down to the “nominal frequency.” If “ΣPGEN (NR) 1 ΣPGEN n (R)” is greater than “ΣPLOAD n” then the frequency will settle down to a value higher than the “nominal frequency.” c. The “reference frequency” for both the “dependency of the load on frequency” (“load damping”) term and the “governor response” term The term “k1 fn” in Eq. (8.1a) contains the “reference frequency” for both the “dependency of the load on frequency” (“load damping”) term and the “governor response” term. In this instance the nominal frequency fn is being applied to both. This does not necessarily have to be so since the synchronous generator governor response could have a reference or “target” frequency different from the nominal frequency. It is simpler to do it like this, though, for the purposes of analysis. However, the “load damping” term is invariably based on the nominal frequency fn. d. The “instantaneous trip” term The term “ 2 PTRIP 3 u(t 2 tTRIP)” in Eq. (8.3e) represents an “instantaneous trip” of generation. It can be made easily to represent an “instantaneous trip” of load by simply changing the sign from negative to positive. This term works as follows: when t # tTRIP the term 5 0; when t . tTRIP the term 5 2PTRIP. e. The “power ramp” term The term “a0 1 a1 t” in Eq. (8.3e) represents a kind of general “power ramp.” It can be used to model an increase or decrease in either, or both, of generation or demand by the appropriate choice of values for the parameters “a0” and “a1.” It is a very general representation of any response
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profile that is “piecewise linear.” It can represent the responsive output of a synchronous generator during the “primary response” phase or perhaps the initial output of a general responsive device (e.g., a battery).
8.3.4 A note on the application of the “state-space” formulation to general “stability and control” problems The earlier “state-space” representation, being equivalent in effect to an algebraic-differential representation, can be used for modeling and finding solutions to other frequency stability and control problems (please see, for example, Ref. [1], Chapter 12 for an application to small-signal stability analysis). We recall that it is being employed here in this book as a kind of stepping-stone to the “transfer function” format version we wish to set up for our version of the “swing equation.”
8.4 Conversion from the “state-space” format to the “transfer function” format 8.4.1
An introduction to the “transfer function” format
There is a standard procedure by which an equation in the “state-space” format can be converted in a straightforward, although, in some cases, perhaps arduous, manner to an equivalent representation in the “transfer function” format (for example, please see Ref. [2]. p. 761 for more details). We now proceed to accomplish this conversion for our case of the “swing equation.”
8.4.2 Converting the “state-space” format of the “swing equation” given in Eq. (8.3) to “transfer function” format Now that we have obtained the “swing equation” in the “state-space” format in the form of Eqs. (8.3a)(8.3f) inclusive, we are now ready to convert this representation to the “transfer function” format. To accomplish this task, we are going to apply the formula in equation (11-2) on p. 753 of Ref. [2] to Eq. (8.3a). Having done this, the resulting expression for the output Y(s) divided by the input U(s), using the same terminology as in Eq. (8.3a), is Y ðsÞ b1 5 U ðsÞ a0 s 1 a1
ð8:4Þ
Y ð sÞ 1 5 U ðsÞ ð2Hf0 Us 1 k1 Þ
ð8:5Þ
or
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In calculating which we have also used the results from Eqs. (8.3b) to (8.3d).
8.4.3 The “gain” and “time constant” associated with the system described by Eq. (8.3) Two further important pieces of information contained within Eq. (8.3) are the “gain K” and “time constant T.” These can be deduced in a straightforward manner from Eq. (8.4). If we introduce the “gain K” and the “time constant T” by adopting the standard formula: Y ð sÞ K 5 U ðsÞ 1 1 sUT
ð8:6Þ
then by a straight comparison of Eq. (8.6) with Eq. (8.5) we therefore find that K5
1 k1
ð8:7aÞ
and T 5 2UHU
f0 k1
ð8:7bÞ
or, equivalently, K5
1 a1
ð8:8aÞ
and T 5 2UHU
f0 a1
ð8:8bÞ
by using the result from Eq. (8.3c). Please see Ref. [1], p. 586 for further details on this topic.
8.5 Conversion from the “transfer function” format to the “control diagram” format 8.5.1
An introduction to the “control diagram” format
The “control diagram” format is simply the familiar “block diagram” that we can now construct easily, since we now know the details of the “transfer function” as given by Eq. (8.6).
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Δu = Δ(PG – PL)–
1 . 2Hf0.s + k1
Δy = Δ(f – fn)
DIAGRAM 8.1 A simple control (block) diagram representation of the “standard swing equation” [Eqs. (8.1a)(8.1d)].
Δy = Δ(f – f0)
1/2H.f0.s
Δu = Δ(PG – PL)
k1 DIAGRAM 8.2 An alternative control diagram representation of the “standard swing equation” [Eqs. (8.1a)(8.1d)].
8.5.2 Converting the “transfer function” representation of the “standard swing equation” of Eq. (8.5) to “control diagram” format From Eq. (8.5), we may now draw the corresponding “control (block) diagram” as shown in Diagram 8.1. We see graphically from Diagram 8.1 how a mismatch between system generation and system demand (load) translates by means of a “transfer function” expressed in Laplace transform format into the deviation of the power frequency from the nominal power frequency. Note that in Diagram 8.1 the input variable u 5 “PG 2 PL” is given by Eq. (8.3e), and the output variable y 5 f 2 fn 5 Δf is given by Eq. (8.2c).
8.5.3 An alternative “control diagram” format for the “standard swing equation” [Eqs. (8.1a)(8.1d)] A second way to represent the “swing equation” in “control diagram” format is to separate away the “k1” term in Diagram 8.1 into a “feedback” loop, as in Diagram 8.2. where again u 5 “PG 2 PL” is given by Eq. (8.3e), and y 5 f 2 fn 5 Δf is given by Eq. (8.2c).
8.5.4
A comparison of Diagram 8.2 with Diagram 8.1
1. The equivalence of Diagrams 8.1 and 8.2 are equivalent to each other.
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It is left as an exercise to the interested reader to prove that Diagrams 8.1 and 8.2 are logically equivalent to one another. [Hint: Let the “feedforward branch” be called G and let the “feedback branch” be called H.] 2. The advantage of Diagram 8.2 over Diagram 8.1 The advantage of the representation shown in Diagram 8.2 over the representation shown in Diagram 8.1 is that Diagram 8.2 clearly shows that the term “k1” is a simple “feedback” term, whereas this fact is not as clear from looking at Diagram 8.1.
8.6 Iterating the values in the “control diagram” representation of the “swing equation” to find a solution for the “system frequency” as a function of time 8.6.1
The overall objective of the process
The overall objective of this process is to find out, by calculation, how much generation needs to be held in reserve (the “response requirement”) to ensure that for a given loss of generation the ensuing path of the frequency does not transgress the statutory lower limit for the system frequency during the “primary response” period. This period extends from the initial loss about 10 seconds following the loss, although this timescale will vary from system to system.
8.6.2
The “initial condition” in the “control diagram” method
1. An introduction to the “initial condition” of the “control diagram” method The “initial condition” of the “control diagram” method may clearly be either a “steady state” or “not a steady state.” We shall consider these two possibilities in turn: a. If the “initial condition” of the “control diagram” method is a “steady state” If the “initial condition” of the “control diagram” is a “steady state,” then, by definition, at t 5 0, at whatever frequency the system is running at, the system generation and demand are exactly in balance at that frequency, and the frequency therefore is momentarily unchanging. Further, if nothing on the system changes to disturb this equilibrium, the system frequency will stay constant at its current value, whatever that happens to be. b. If the “initial condition” of the “control diagram” method is “not a steady state” If the “initial condition” of the “control diagram” is “not a steady state,” then, by definition, at t 5 0, at whatever frequency, the system generation and demand are not in balance. Therefore the system frequency will change to some other value.
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How the “control diagram” reacts to the status of the “input”
1. If the “input u” is in a “steady state” If the “input u” is in a “steady state” with Δu 5 0 (that is, the system “generation” and “demand” are “in balance” with no change), then the “output y” will not change, that is, Δy 5 0. 2. If the “input u” is not in a “steady state” If the “input u” is not in a “steady state” with Δu6¼0 (that is, the system generation and demand are not “in balance”), then the “output y” will change, that is, Δy6¼0 as well.
8.6.4 How to perform a “single iteration” using the “control diagram” method 1. The meaning of an iteration in the “control diagram” method. What we mean by an “iteration” in the “control diagram” method is as follows: a. We start with the initial values for generation and demand. b. We remove an amount of generation or demand to represent a “trip.” c. We increment or decrement as appropriate the response by an amount “Δu” according to an estimated value of the amount of “frequency response” that can be made available in one step-length Δt. d. We apply the iteration procedure as described in the next paragraph to find the change in the frequency “Δf” corresponding to our applied change to the input “Δu.” 2. The iteration procedure To perform a “single iteration,” or “step,” using the “control diagram” representation of the “swing equation,” as given in Diagram 8.1, we must go through the following process (please see Ref. [1], pp. 585587 for an example of how to follow this process): a. Calculate “Δu” in “Laplace transform space,” that is, find “Δu(s)” To calculate “Δu(s)” it is necessary to take the input power balance (which may or may not be zero) and divide it by s (Ref.[1], p. 586). b. Calculate “Δy” in “Laplace transform space,” that is, find “Δy(s)” To calculate “Δy(s)” we take the input power balance “Δu” divided by s and then multiply it by the “transfer function” K/(1 1 sT) as defined earlier. c. Calculate the “inverse Laplace transform” of “Δy(s)” to find “Δy(t)” The final step of this single-step process is to calculate the “inverse Laplace transform” of “Δy(s)” obtained as described earlier to find “Δy(t).” d. Choose the time step Choose the time step Δt, for example, 0.05 seconds. e. Plotting the result for the single iteration Finally, we plot “Δy(t)” against “t” for one time step Δt.
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8.6.5 How to perform “multiple iterations” of the “control diagram” method 1. Repetition of the “single iteration” To achieve “multiple iterations” of the “control diagram” method, we repeat the process described in Section 8.6.4 with updated values for the input “Δu.” The updated value is the value from the previous iteration plus an increment calculated according to an assumed “responsive generator ramp rate R.” This is an assumed amount in MW/s calculated according to the ramp rate for a single responsive machine of a selected size multiplied by the number of responsive machines that we estimate to be available. 2. Adjusting the amount of response available Once a full plot has been drawn up to a selected time (say 10 seconds), the plot is examined to determine whether the frequency is within the required bounds for all values of time within the plotting range. If the frequency is within the required bounds for all values of time within the plotting range, the process is terminated. If the frequency is not within the required bounds for all values of time within the plotting range, then the amount of frequency response available is updated, and the whole process repeated until the frequency is within the required bounds. 3. The “convergence” of the calculation process The process described in Section 8.6.3 is repeated until “convergence” is attained by some definition given in the preceding paragraph. 4. Acceptable frequency behavior To find the “frequency response requirements” for a system the “response provision” that can be included with the “input Δu” is varied until the resulting frequency behavior f(t) becomes acceptable, that is, the “system frequency” does not rise too high or fall too low in a certain timescale that is defined by the relevant “operational standards” and within preselected tolerances. 5. Choosing “convergence tolerances” In our example a “convergence tolerance” for Δu or Δf(t) (or both) can be defined such that when the change resulting from an iteration is smaller than one or both tolerances, the iterations are terminated. We can then say that we have “the solution” for the “response provision” required.
8.7 MATLAB example simulation 8: an example of the “control diagram” method: modeling a generator “ramped response” 8.7.1
Introduction
In this exercise, we shall be following the prescription laid out in Section 8.6.4 of this chapter to solve an example problem in which an initial
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power balance between generation and demand on a system is disturbed by a loss of generation from the system caused by an unexpected “trip.” We shall assume that there is a source of ramped response available which attempts to restrain the ensuing fall in frequency and to return the “system frequency” as far as possible to normal operating levels. This ramped response is represented by the two terms a0 1 a1t using our standard notation. At the end of our calculations, we shall create a plot of the “system frequency” f versus time t to show the time-evolution of the “system frequency.”
8.7.2
A statement of the problem
Let us adopt the system of MATLAB example simulation 1 which is a small 50 Hz system with a demand/load of 3000 MW. Initially 10 identical generators, each with an “inertia constant” H 5 5.0 seconds are running at their rated output of 300 MW each. At t 5 0 second one of the generators trips off the system. We need to calculate the time-evolution of the “system frequency” up to a time t 5 10 seconds.
8.7.3
The general solution of the problem
The general solution of the problem is as described in Sections 8.6.4 and 8.6.5. Section 8.6.4 explains a single iteration of the process, while Section 8.6.5 explains multiple iterations.
8.7.4
The input data
1. General data We need to feed data to represent Eqs. (8.1a) and (8.3f) with k 5 0 into the program. We list these two following equations again here for ease of reference: Σ PGEN ðNRÞ 1 Σ PGEN n ðRÞ 2 Σ PLOAD n 2 k1 Uðf fn Þ 2 PTRIP Uuðt tTRIP Þ df 1 a0 1 a1 t 5 2UHUf0 U dt ð8:9Þ k1 5 ΣPLOAD n Uα 1 ΣPGEN n ðRÞ Uk
ð8:10Þ
where as noted earlier, we shall take k 5 0 because we are not modeling a generator governor response, but ramped response from some other (electronic) source using the combination of parameters a0 1 a1 t to describe the ramp.
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With reference to Eqs. (8.9) and (8.10) the data we shall need is as follows: ΣPGEN ðNRÞ 53000 MW; ΣPGEN n ðRÞ 5 0 MW; fn 5 50 Hz; PTRIP 5 300 MW; tTRIP 5 0 s; H 5 5:0 s; ΣPLOAD n 5 3000 MW; α 5 0:02 pu=Hz: 2. Data for the responsive ramp We need to choose some values for the “ramp parameters” a0 and a1 to model some provision of response following the generation loss. Let us assume for this example that the response is instantaneous, that is, there is no time delay between the generator trip and the start of the responsive ramp. In this limiting case we have a0 5 0
ð8:11Þ
Let us also assume that we have 150 MW of a responsive source available which can change its output from 0 to 150 MW in 5 seconds. In this case we have a1 5 30 MW=s
ð8:12Þ
for our second “ramp parameter.” 3. Calculation of the input parameter u Since we are framing the problem in terms of the response of an output y to an input u, we need to begin by finding values for the input variable u. This variable is given by Eq. (8.3e). We repeat this following equation here for ease of reference: u 5 ΣPGEN ðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n 2 PTRIP Uuðt tTRIP Þ 1 a0 1 a1 t ð8:13Þ This expression may now be evaluated by substituting the data from Section 8.7.4 (1) into Eq. (8.13). Doing this, we get u 5 3000 MW 1 0 MW 3000 MW 300UuðtÞ 1 0 1 30 MW=s Ut or
u 5 2 300UuðtÞ 1 30 MW=s Ut
ð8:14Þ
4. Choosing the “time-step” Δt Let us divide the ramping-up time of 5 seconds by 100 to give a time step of Δt 5 0:05 s
ð8:15Þ
We shall use this time-step to increment the input u and hence the output y.
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The output data
The output data will consist of values of Δy 5 Δ(f 2 fn) that correspond to increments of the input Δu 5 Δ(PG 2 PL) as shown in Diagrams 8.1 and 8.2. The output values for each step are obtained by following the procedure described in Sections 8.6.4 and 8.6.5. Values of f 2 fn can then be plotted against time t. Alternatively, values of f may be plotted against t.
8.7.6
The suggested MATLAB code
The suggested MATLAB code to solve the problem is as follows: % % % %
Solving the “Swing Equation” by the “control diagram” method. Enter the input data in physical units: PNR_P 5 input(‘Please enter the total output of the nonresponsive generation in MW’) PR_P 5 input(‘Please enter the total rated output of the responsive generation In MW’) FN_P 5 input(‘Please enter the system nominal frequency in Hz’) PTRIP_P 5 input(‘Please enter the amount of generation tripped in MW’) TTRIP 5 input(‘Please enter the time of the generation trip in seconds’) H 5 input(‘Please enter the machine inertia constant in seconds’) PLOAD_P 5 input(‘Please enter the total system load in MW’) ALPHA_P 5 input(‘Please enter the “load damping” constant α in per cent per Hz’)
% % %
Now enter the “responsive ramp” data: A0_P 5 input(‘Please enter the value of the “ramp” constant a0in MW’) A1_P 5 input(‘Please enter the value of the “ramp” constant a1in MW per second)
% %
Convert the data as necessary to “per unit” by defining the MVA and f bases:
% MVA_BASE 5 3000/0.8 F_BASE 5 FN_P % %
Now convert all quantities to “per unit” as necessary:
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% PNR 5 PNR_P/MVA_BASE PR 5 PR_P/MVA_BASE FN 5 FN_P/F_BASE PTRIP 5 PTRIP_P/MVA_BASE PLOAD 5 PLOAD_P/MVA_BASE ALPHA 5 ALPHA_P F_BASE/100 A0 5 A0_P/MVA_BASE A1 5 A1_P/MVA_BASE % % %
Now enter the size of the time step in seconds: TSTEP 5 input(‘Please enter the value of the time-step in seconds’)
% % % %
Calculate the values of the parameters K and T of the transfer function using (Eq 8.8a) and (Eq 8.8b): K 5 1/A1 T 5 2 H F0/A1
% % % % % %
In Laplace space Δu(s) 5 Δu/s. We need to multiply this by K/(1 1 s.T) to get Δy(s), of which we can then take the inverse transform to get Δy(t). For the first time increment only: DELTA _OMEGA(1) 5 PTRIP K (EXP(-TSTEP/T) 1) DELTA_F(1) 5 2 PI DELTA_OMEGA(1)
% % % %
Now perform iterations to generate the successive speed changes up to a time limit of 5 seconds: NITER 5 5/TSTEP
% for k 5 2, NITER T 5 k TSTEP DELTA_OMEGA(k) 5 A1 T K (EXP(-TSTEP/T) 1) DELTA_F(k) 5 2 PI DELTA_OMEGA(k) end % %
A graph of the results may now be printed out.
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Instructions for the user
Enter the required input data when prompted by the MATLAB code. The output will be a graph of “system frequency” against time for time t 5 010 seconds.
8.8 Comparison of the “control diagram” approach with some of the other methods available 8.8.1
Introduction
The “control diagram” approach is a sort of hybrid between the “analytical methods” and the “numerical methods”: it contains elements of both. We describe these similarities in turn next.
8.8.2 Similarity of the “control diagram” approach to the “analytical methods” The “control diagram” approach has similarities with the “analytical methods.” There are elements of the “Laplace transform” method because Laplace transforms are needed as part of the “control diagram” approach as well to calculate the change in frequency output as a response to a change in the power balance input.
8.8.3 Similarity of the “control diagram” approach to the “numerical methods” There is also an element of the “numerical methods” in the “control diagram” approach because iterations are required in both methods to find the solution.
8.8.4
Uniqueness of the “control diagram” approach
The “control diagram” approach is unlike the other two groups of methods in its use of the “block diagram” and in the “state-space” approach.
8.9 The next chapter: some important practical applications 8.9.1
Introduction
In the next chapter, we look at some important practical applications. They are as follows: G
G
Calculation of the “rate of change of frequency” (RoCoF) following a loss of generation or demand Calculation of the “system response requirements” following a loss of generation (the “low frequency excursion”)
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Calculation of the “system response requirements” following a loss of demand (the “high frequency excursion”) Calculation of the “system response requirements” during “normal operation” (relatively small variations in the “system frequency”)
These calculations are all important tasks performed regularly every day by control engineers at national control centers around the world. Because of their importance, we give short descriptions of each in preparation for the next chapter as follows:
8.9.2 [I] The calculation of the “rate of change of frequency” following a loss of generation or demand 1. RoCoF relays a. The equipping of generators with “RoCoF relays” Some generators are equipped with “RoCoF relays”. b. Why are “RoCoF” relays installed on generators? “RoCoF” relays may be installed on generators for several reasons: i. Loss of grid/loss of mains/islanding protection One potential use of having “RoCoF” relays on generators is to detect a “loss of grid” condition (also known as “loss of mains” or “islanding”). The purpose of the “RoCoF” relays in this situation is to protect the generator. When the “RoCoF” relay detects a “loss of grid” condition, the generator is “tripped,” that is, disconnected from the main system. ii. The detection of “under-frequency” and “over-frequency” conditions “RoCoF relays” may also be installed on generators to detect both “under-frequency” and “over-frequency” conditions. The purpose of the “RoCoF relay” here is to protect the generator by tripping it before the frequency becomes too low or too high, thus avoiding potentially costly damage to the generator. 2. The control of the RoCoF The “RoCoF” on the power system can be controlled by the scheduling of generation to ensure that there is enough inertia on the system to limit the RoCoF to below generator tripping threshold levels. 3. Calculating the “RoCoF” Calculating the “RoCoF” is achieved straightforwardly by evaluating the right-hand side of the “swing equation,” since “RoCoF” is simply “df/dt.”
8.9.3 [II] The calculation of the “system response requirements” following a loss of generation (a “low frequency excursion”) 1. What is a “low frequency excursion”?
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By “low frequency excursion” we mean that the “system frequency” has deviated, perhaps significantly, from the “normal operating range” and is lower than where it started. 2. Calculating the “system response requirements” following a loss of generation (a “low frequency excursion”) Calculating the “system response requirements” following a loss of generation (a “low frequency excursion”) may be achieved by using the “swing equation” to model sources of generation and demand including “frequency responsive plant” and iterating one of the solution methods in Chapters 68 following a loss of generation to find the amount of “frequency response” needed to maintain the “system frequency” to above the low voltage tolerance required by the “statutory requirements” for that size of generation loss.
8.9.4 [III] The calculation of the “system response requirements” following a loss of demand (a “high frequency excursion”) 1. What is a “high frequency excursion”? By “high frequency excursion” we mean that the “system frequency” has deviated, perhaps significantly, from the “normal operating range” and is higher than where it started. 2. Calculating the “system response requirements” following a loss of demand (a “high frequency excursion”) Calculating the “system response requirements” following a loss of demand (a “high frequency excursion”) may be achieved by using the “swing equation” to model sources of generation and demand including “frequency responsive plant” and iterating one of the solution methods in Chapters 68 following a loss of demand to find the amount of “frequency response” needed to maintain the “system frequency” to below the high voltage tolerance required by the “statutory requirements” for that size of demand loss.
8.9.5 [IV] The calculation of the “system response requirements” during “normal operation” (“relatively small variations” in the “system frequency”) 1. What is “normal operation”? By “normal operation” we mean that the “system frequency” is contained within a narrow band around the “nominal frequency.” There are no “low frequency excursions” or “high frequency excursions.” 2. Calculating the “system response requirements” during “normal operation” (relatively small variations in the “system frequency”) Calculating the “system response requirements” during “normal operation” (relatively small variations in the “system frequency”) may be
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achieved by using the “swing equation” to model sources of generation and demand including “frequency responsive plant” and iterating one of the solution methods in Chapters 68 following a selected small size of loss of generation or demand to find the amount of “frequency response” needed to maintain the “system frequency” both above the low voltage tolerance and below the high voltage tolerance defined by the “normal band of operation” around the “nominal frequency.”
References Books [1] P. Kundur, 22nd reprint Power System Stability and Control. Power System Engineering Series, Electric Power Research Institute, McGraw-Hill Education, 2017. [2] K. Ogata, Modern Control Engineering, fourth ed., Prentice Hall/Pearson Education International, 2002. [3] D.R. Towill, Transfer Function Techniques for Control Engineers, Iliffe Books Ltd, 1970. [4] J.G. Truxal, Control Engineers’ Handbook, McGraw-Hill Book Company, Inc., 1958.
The “control diagram” method 8.1 A short introduction to the “control diagram” method 8.1.1
A third approach to solving the “Swing Equation”
In addition to the “exact solution of the differential equation” and “Laplace transform” methods covered in Chapter 6, Analytical solutions, and the “numerical solution” methods of solving the “swing equation” covered in Chapter 7, Numerical methods for solving the “swing equation,” of this book, there is a third possible approach available, which may be termed the “control diagram” method. This approach couches the problem of the solution of the “swing equation” in the language of “control engineering” by using, instead of algebraic and differential equations, “block diagrams” or “control diagrams.” For examples of this approach, please see Refs. [3] and [4] listed at the end of this chapter. The problem is the same as for the other methods, namely solving Eq. (7.1), but the way the solution is approached is slightly different from that of the other methods.
8.1.2 The use of the “control diagram” to represent a differential equation The idea of using a “control diagram” to represent a differential equation such as the “swing equation” is, of course, by no means new (please see, for example, Ref. [1], Figures 11.4 and 11.5 on p. 585 for a basic illustration of the “control diagram” representation of the elementary “swing equation” with the addition of a single feedback loop to model the effect of “load damping,” that is, the variation of load with frequency). However, it can be a useful and visually appealing means of better understanding the relationships between the various physical and logical elements in an algebraic-differential equation such as the “swing equation.” For example, it groups the sum of all the sources of generation minus the demand into one element: the input, and the change in frequency as the output. In this way it separates out “cause” and “effect” by representing the system frequency equations as “input” and “output,” and it represents the damping-type terms all together as one feedback loop. Modern Aspects of Power System Frequency Stability and Control. DOI: https://doi.org/10.1016/B978-0-12-816139-5.00008-4 © 2019 Elsevier Inc. All rights reserved.
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Development of the “control diagram” representation
The basic representation of the control diagram given in Ref. [1], Figures 11.4 and 11.5 on p. 585 can be extended to include the extra features contained in our “standard version” of the “swing equation” given in Eqs. (8.1a)(8.1d). Thus in addition to the features contained already in Figures 11.4 and 11.5 on p. 585 of Ref. [1], namely the system power balance including the variation of load with frequency (load damping) and generator “governor response,” we can also include further elements, such as the “general responsive profile”, represented in Eqs. (8.1a)(8.1d) by a0 1 a1 t. However, representing the unit step-change term PTRIP u(t 2 tTRIP) of Eqs. (8.1a)(8.1d) may prove more challenging using the “control system” method.
8.2 Recalling the “swing equation” 8.2.1
The “standard form” of the “swing equation”
Let us quote, for convenience, and for further use in the present chapter, the “standard form” of the “swing equation” that was last given in the previous chapter as [Eq. (7.1)]: ΣPGENðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n 2 k1 Uðf fn Þ 2 PTRIP Uuðt tTRIP Þ df 1 a0 1 a1 t 5 2UHUf0 U dt ð8:1aÞ
8.2.2
A simplifying substitution
We can simplify [Eq. (8.1a)] further for our purposes in this chapter by making the substitution: Δf f fn
ð8:1bÞ
so that Eq. (8.1a) may be rewritten in a more convenient form as 2UHUf0 U
dðΔf Þ 1 k1 UðΔf Þ 5 ΣPGEN ðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n dt
ð8:1cÞ
2 PTRIP Uuðt tTRIP Þ 1 a0 1 a1 t where in addition we have taken advantage of the fact that dðΔf Þ df 5 dt dt which is true since we have dfn/dt 5 0.
ð8:1dÞ
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All that we have done here is to change the dependent variable from f to Δf because this fits in better with the “block diagram” representation we are going to use.
8.2.3
In the next subsection
In the next subsection we are going to start to look at how to formulate the “control system” method, beginning with Eq. (8.1c). This involves first representing the “swing equation” in “state-space format.” This is an intermediate step between the algebraic-differential representation we are starting with and the “control diagram” or “block diagram” representation we are going to finish up with.
8.3 The representation of the “swing equation” in “state-space” format 8.3.1
An introduction to “state-space” format
First, the equation [Eqs. (8.1a)(8.1d)] can be expressed, in a completely equivalent way, in the following well-known “state-space” format, which may be defined as follows: x_ 5 AUx 1 BUu
ð8:2aÞ
y 5 CUx 1 DUu
ð8:2bÞ
and
(see, for example, Ref. [1], p. 704 for a more detailed description of this representation). In Eqs. (8.2a) and (8.2b) the variable u is the “input variable,” the variable y is the “output variable,” and the variable x is the “system variable” or “state variable.” The parameters A, B, C, and D are constants. The “state-space” format has the in-built flexibility for choosing the variables and constants to suit a desired application—a flexibility that we shall be making use of in the next section.
8.3.2 Application of the “state-space” format to the “swing equation” To represent the “swing equation,” in the form represented by Eq. (8.1c), in the “state-space” format of Eqs. (8.2a) and (8.2b), we may select x 5 y 5 Δf 5 f fn
ð8:2cÞ
as the output variable. This represents the amount that the “system frequency” differs from the “nominal frequency.”
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Following Ref. [2], p. 753 and using their notation, our version of the “swing equation” given in Eq. (8.1c) may be written in the form: a0 Uy_ 1 a1 Uy 5 b1 Uu
ð8:3aÞ
so that, in their notation, we have a0 5 2UHUf0
ð8:3bÞ
a1 5 k 1
ð8:3cÞ
b1 5 1
ð8:3dÞ
and we may take
For the input variable u we therefore have u 5 ΣPGEN ðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n 2 PTRIP Uuðt tTRIP Þ 1 a0 1 a1 t ð8:3eÞ Thus we have provided values for the three parameters A, B, and u of Eq. (8.2a) in addition to taking C 5 1 and D 5 0 in Eq. (8.2b) to complete the conversion of the “swing equation” into “state-space” format. In the next section, we shall provide an interpretation of some of the terms in the above set of equations.
8.3.3 An interpretation of some of the terms in Eqs. (8.3a)(8.3e) inclusive 1. A comment on the information distribution of Eqs. (8.3a)(8.3e) We can see from the list of equations earlier that most of the information for the “state-space” format of the “swing equation” is contained in Eqs. (8.3c) and (8.3e). We now therefore proceed to look more closely at these two equations, taking them in turn. 2. The information contained in Eq. (8.3c) In Eq. (8.3c), there is information contained about the dependency of the “system frequency” f on itself since it multiplies y in Eq. (8.3a). It is therefore a kind of “feedback” term. We shall see that this assertion will be confirmed graphically later in our analysis. We recall from previous chapters, for example, in Eq. (7.1a), that the parameter k1 is given by k1 5 ΣPLOAD n Uα 1 ΣPGEN n ðRÞ Uk
ð8:3fÞ
where ΣPLOAD n represents the total MW load at nominal frequency, α is the “load-damping coefficient,” ΣPGEN n (R) represents the total MW generation of responsive synchronous machines at nominal frequency, and k is a parameter essentially equal to the reciprocal of the slope D of
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the synchronous machine governor characteristic [see, for example, Eq. (A.9) of Appendix A for the full definition of this parameter]. 3. The information contained in Eq. (8.3e) In Eq. (8.3e), we have all the information relating to the net input to the system: generation minus demand (or load: demand and load are the same here because we are ignoring I2R losses). We can divide Eq. (8.3e) into the following groups of terms: a. The basic “power balance” term at “nominal frequency” The term “ΣPGEN (NR) 1 ΣPGEN n (R) 2 ΣPLOAD n” in Eq. (8.3e) represents the basic power balance at the “nominal frequency” of the system in the absence of any “trips” and before any “responsive output” has appeared on the system. b. In the absence of any responsive output only If “ΣPGEN (NR) 1 ΣPGEN n (R)” is less than “ΣPLOAD n” then the frequency will settle down to a value lower than the “nominal frequency.” If “ΣPGEN (NR) 1 ΣPGEN n (R)” equals “ΣPLOAD n” then the frequency will settle down to the “nominal frequency.” If “ΣPGEN (NR) 1 ΣPGEN n (R)” is greater than “ΣPLOAD n” then the frequency will settle down to a value higher than the “nominal frequency.” c. The “reference frequency” for both the “dependency of the load on frequency” (“load damping”) term and the “governor response” term The term “k1 fn” in Eq. (8.1a) contains the “reference frequency” for both the “dependency of the load on frequency” (“load damping”) term and the “governor response” term. In this instance the nominal frequency fn is being applied to both. This does not necessarily have to be so since the synchronous generator governor response could have a reference or “target” frequency different from the nominal frequency. It is simpler to do it like this, though, for the purposes of analysis. However, the “load damping” term is invariably based on the nominal frequency fn. d. The “instantaneous trip” term The term “ 2 PTRIP 3 u(t 2 tTRIP)” in Eq. (8.3e) represents an “instantaneous trip” of generation. It can be made easily to represent an “instantaneous trip” of load by simply changing the sign from negative to positive. This term works as follows: when t # tTRIP the term 5 0; when t . tTRIP the term 5 2PTRIP. e. The “power ramp” term The term “a0 1 a1 t” in Eq. (8.3e) represents a kind of general “power ramp.” It can be used to model an increase or decrease in either, or both, of generation or demand by the appropriate choice of values for the parameters “a0” and “a1.” It is a very general representation of any response
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profile that is “piecewise linear.” It can represent the responsive output of a synchronous generator during the “primary response” phase or perhaps the initial output of a general responsive device (e.g., a battery).
8.3.4 A note on the application of the “state-space” formulation to general “stability and control” problems The earlier “state-space” representation, being equivalent in effect to an algebraic-differential representation, can be used for modeling and finding solutions to other frequency stability and control problems (please see, for example, Ref. [1], Chapter 12 for an application to small-signal stability analysis). We recall that it is being employed here in this book as a kind of stepping-stone to the “transfer function” format version we wish to set up for our version of the “swing equation.”
8.4 Conversion from the “state-space” format to the “transfer function” format 8.4.1
An introduction to the “transfer function” format
There is a standard procedure by which an equation in the “state-space” format can be converted in a straightforward, although, in some cases, perhaps arduous, manner to an equivalent representation in the “transfer function” format (for example, please see Ref. [2]. p. 761 for more details). We now proceed to accomplish this conversion for our case of the “swing equation.”
8.4.2 Converting the “state-space” format of the “swing equation” given in Eq. (8.3) to “transfer function” format Now that we have obtained the “swing equation” in the “state-space” format in the form of Eqs. (8.3a)(8.3f) inclusive, we are now ready to convert this representation to the “transfer function” format. To accomplish this task, we are going to apply the formula in equation (11-2) on p. 753 of Ref. [2] to Eq. (8.3a). Having done this, the resulting expression for the output Y(s) divided by the input U(s), using the same terminology as in Eq. (8.3a), is Y ðsÞ b1 5 U ðsÞ a0 s 1 a1
ð8:4Þ
Y ð sÞ 1 5 U ðsÞ ð2Hf0 Us 1 k1 Þ
ð8:5Þ
or
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In calculating which we have also used the results from Eqs. (8.3b) to (8.3d).
8.4.3 The “gain” and “time constant” associated with the system described by Eq. (8.3) Two further important pieces of information contained within Eq. (8.3) are the “gain K” and “time constant T.” These can be deduced in a straightforward manner from Eq. (8.4). If we introduce the “gain K” and the “time constant T” by adopting the standard formula: Y ð sÞ K 5 U ðsÞ 1 1 sUT
ð8:6Þ
then by a straight comparison of Eq. (8.6) with Eq. (8.5) we therefore find that K5
1 k1
ð8:7aÞ
and T 5 2UHU
f0 k1
ð8:7bÞ
or, equivalently, K5
1 a1
ð8:8aÞ
and T 5 2UHU
f0 a1
ð8:8bÞ
by using the result from Eq. (8.3c). Please see Ref. [1], p. 586 for further details on this topic.
8.5 Conversion from the “transfer function” format to the “control diagram” format 8.5.1
An introduction to the “control diagram” format
The “control diagram” format is simply the familiar “block diagram” that we can now construct easily, since we now know the details of the “transfer function” as given by Eq. (8.6).
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Δu = Δ(PG – PL)–
1 . 2Hf0.s + k1
Δy = Δ(f – fn)
DIAGRAM 8.1 A simple control (block) diagram representation of the “standard swing equation” [Eqs. (8.1a)(8.1d)].
Δy = Δ(f – f0)
1/2H.f0.s
Δu = Δ(PG – PL)
k1 DIAGRAM 8.2 An alternative control diagram representation of the “standard swing equation” [Eqs. (8.1a)(8.1d)].
8.5.2 Converting the “transfer function” representation of the “standard swing equation” of Eq. (8.5) to “control diagram” format From Eq. (8.5), we may now draw the corresponding “control (block) diagram” as shown in Diagram 8.1. We see graphically from Diagram 8.1 how a mismatch between system generation and system demand (load) translates by means of a “transfer function” expressed in Laplace transform format into the deviation of the power frequency from the nominal power frequency. Note that in Diagram 8.1 the input variable u 5 “PG 2 PL” is given by Eq. (8.3e), and the output variable y 5 f 2 fn 5 Δf is given by Eq. (8.2c).
8.5.3 An alternative “control diagram” format for the “standard swing equation” [Eqs. (8.1a)(8.1d)] A second way to represent the “swing equation” in “control diagram” format is to separate away the “k1” term in Diagram 8.1 into a “feedback” loop, as in Diagram 8.2. where again u 5 “PG 2 PL” is given by Eq. (8.3e), and y 5 f 2 fn 5 Δf is given by Eq. (8.2c).
8.5.4
A comparison of Diagram 8.2 with Diagram 8.1
1. The equivalence of Diagrams 8.1 and 8.2 are equivalent to each other.
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It is left as an exercise to the interested reader to prove that Diagrams 8.1 and 8.2 are logically equivalent to one another. [Hint: Let the “feedforward branch” be called G and let the “feedback branch” be called H.] 2. The advantage of Diagram 8.2 over Diagram 8.1 The advantage of the representation shown in Diagram 8.2 over the representation shown in Diagram 8.1 is that Diagram 8.2 clearly shows that the term “k1” is a simple “feedback” term, whereas this fact is not as clear from looking at Diagram 8.1.
8.6 Iterating the values in the “control diagram” representation of the “swing equation” to find a solution for the “system frequency” as a function of time 8.6.1
The overall objective of the process
The overall objective of this process is to find out, by calculation, how much generation needs to be held in reserve (the “response requirement”) to ensure that for a given loss of generation the ensuing path of the frequency does not transgress the statutory lower limit for the system frequency during the “primary response” period. This period extends from the initial loss about 10 seconds following the loss, although this timescale will vary from system to system.
8.6.2
The “initial condition” in the “control diagram” method
1. An introduction to the “initial condition” of the “control diagram” method The “initial condition” of the “control diagram” method may clearly be either a “steady state” or “not a steady state.” We shall consider these two possibilities in turn: a. If the “initial condition” of the “control diagram” method is a “steady state” If the “initial condition” of the “control diagram” is a “steady state,” then, by definition, at t 5 0, at whatever frequency the system is running at, the system generation and demand are exactly in balance at that frequency, and the frequency therefore is momentarily unchanging. Further, if nothing on the system changes to disturb this equilibrium, the system frequency will stay constant at its current value, whatever that happens to be. b. If the “initial condition” of the “control diagram” method is “not a steady state” If the “initial condition” of the “control diagram” is “not a steady state,” then, by definition, at t 5 0, at whatever frequency, the system generation and demand are not in balance. Therefore the system frequency will change to some other value.
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How the “control diagram” reacts to the status of the “input”
1. If the “input u” is in a “steady state” If the “input u” is in a “steady state” with Δu 5 0 (that is, the system “generation” and “demand” are “in balance” with no change), then the “output y” will not change, that is, Δy 5 0. 2. If the “input u” is not in a “steady state” If the “input u” is not in a “steady state” with Δu6¼0 (that is, the system generation and demand are not “in balance”), then the “output y” will change, that is, Δy6¼0 as well.
8.6.4 How to perform a “single iteration” using the “control diagram” method 1. The meaning of an iteration in the “control diagram” method. What we mean by an “iteration” in the “control diagram” method is as follows: a. We start with the initial values for generation and demand. b. We remove an amount of generation or demand to represent a “trip.” c. We increment or decrement as appropriate the response by an amount “Δu” according to an estimated value of the amount of “frequency response” that can be made available in one step-length Δt. d. We apply the iteration procedure as described in the next paragraph to find the change in the frequency “Δf” corresponding to our applied change to the input “Δu.” 2. The iteration procedure To perform a “single iteration,” or “step,” using the “control diagram” representation of the “swing equation,” as given in Diagram 8.1, we must go through the following process (please see Ref. [1], pp. 585587 for an example of how to follow this process): a. Calculate “Δu” in “Laplace transform space,” that is, find “Δu(s)” To calculate “Δu(s)” it is necessary to take the input power balance (which may or may not be zero) and divide it by s (Ref.[1], p. 586). b. Calculate “Δy” in “Laplace transform space,” that is, find “Δy(s)” To calculate “Δy(s)” we take the input power balance “Δu” divided by s and then multiply it by the “transfer function” K/(1 1 sT) as defined earlier. c. Calculate the “inverse Laplace transform” of “Δy(s)” to find “Δy(t)” The final step of this single-step process is to calculate the “inverse Laplace transform” of “Δy(s)” obtained as described earlier to find “Δy(t).” d. Choose the time step Choose the time step Δt, for example, 0.05 seconds. e. Plotting the result for the single iteration Finally, we plot “Δy(t)” against “t” for one time step Δt.
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8.6.5 How to perform “multiple iterations” of the “control diagram” method 1. Repetition of the “single iteration” To achieve “multiple iterations” of the “control diagram” method, we repeat the process described in Section 8.6.4 with updated values for the input “Δu.” The updated value is the value from the previous iteration plus an increment calculated according to an assumed “responsive generator ramp rate R.” This is an assumed amount in MW/s calculated according to the ramp rate for a single responsive machine of a selected size multiplied by the number of responsive machines that we estimate to be available. 2. Adjusting the amount of response available Once a full plot has been drawn up to a selected time (say 10 seconds), the plot is examined to determine whether the frequency is within the required bounds for all values of time within the plotting range. If the frequency is within the required bounds for all values of time within the plotting range, the process is terminated. If the frequency is not within the required bounds for all values of time within the plotting range, then the amount of frequency response available is updated, and the whole process repeated until the frequency is within the required bounds. 3. The “convergence” of the calculation process The process described in Section 8.6.3 is repeated until “convergence” is attained by some definition given in the preceding paragraph. 4. Acceptable frequency behavior To find the “frequency response requirements” for a system the “response provision” that can be included with the “input Δu” is varied until the resulting frequency behavior f(t) becomes acceptable, that is, the “system frequency” does not rise too high or fall too low in a certain timescale that is defined by the relevant “operational standards” and within preselected tolerances. 5. Choosing “convergence tolerances” In our example a “convergence tolerance” for Δu or Δf(t) (or both) can be defined such that when the change resulting from an iteration is smaller than one or both tolerances, the iterations are terminated. We can then say that we have “the solution” for the “response provision” required.
8.7 MATLAB example simulation 8: an example of the “control diagram” method: modeling a generator “ramped response” 8.7.1
Introduction
In this exercise, we shall be following the prescription laid out in Section 8.6.4 of this chapter to solve an example problem in which an initial
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power balance between generation and demand on a system is disturbed by a loss of generation from the system caused by an unexpected “trip.” We shall assume that there is a source of ramped response available which attempts to restrain the ensuing fall in frequency and to return the “system frequency” as far as possible to normal operating levels. This ramped response is represented by the two terms a0 1 a1t using our standard notation. At the end of our calculations, we shall create a plot of the “system frequency” f versus time t to show the time-evolution of the “system frequency.”
8.7.2
A statement of the problem
Let us adopt the system of MATLAB example simulation 1 which is a small 50 Hz system with a demand/load of 3000 MW. Initially 10 identical generators, each with an “inertia constant” H 5 5.0 seconds are running at their rated output of 300 MW each. At t 5 0 second one of the generators trips off the system. We need to calculate the time-evolution of the “system frequency” up to a time t 5 10 seconds.
8.7.3
The general solution of the problem
The general solution of the problem is as described in Sections 8.6.4 and 8.6.5. Section 8.6.4 explains a single iteration of the process, while Section 8.6.5 explains multiple iterations.
8.7.4
The input data
1. General data We need to feed data to represent Eqs. (8.1a) and (8.3f) with k 5 0 into the program. We list these two following equations again here for ease of reference: Σ PGEN ðNRÞ 1 Σ PGEN n ðRÞ 2 Σ PLOAD n 2 k1 Uðf fn Þ 2 PTRIP Uuðt tTRIP Þ df 1 a0 1 a1 t 5 2UHUf0 U dt ð8:9Þ k1 5 ΣPLOAD n Uα 1 ΣPGEN n ðRÞ Uk
ð8:10Þ
where as noted earlier, we shall take k 5 0 because we are not modeling a generator governor response, but ramped response from some other (electronic) source using the combination of parameters a0 1 a1 t to describe the ramp.
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With reference to Eqs. (8.9) and (8.10) the data we shall need is as follows: ΣPGEN ðNRÞ 53000 MW; ΣPGEN n ðRÞ 5 0 MW; fn 5 50 Hz; PTRIP 5 300 MW; tTRIP 5 0 s; H 5 5:0 s; ΣPLOAD n 5 3000 MW; α 5 0:02 pu=Hz: 2. Data for the responsive ramp We need to choose some values for the “ramp parameters” a0 and a1 to model some provision of response following the generation loss. Let us assume for this example that the response is instantaneous, that is, there is no time delay between the generator trip and the start of the responsive ramp. In this limiting case we have a0 5 0
ð8:11Þ
Let us also assume that we have 150 MW of a responsive source available which can change its output from 0 to 150 MW in 5 seconds. In this case we have a1 5 30 MW=s
ð8:12Þ
for our second “ramp parameter.” 3. Calculation of the input parameter u Since we are framing the problem in terms of the response of an output y to an input u, we need to begin by finding values for the input variable u. This variable is given by Eq. (8.3e). We repeat this following equation here for ease of reference: u 5 ΣPGEN ðNRÞ 1 ΣPGEN n ðRÞ 2 ΣPLOAD n 2 PTRIP Uuðt tTRIP Þ 1 a0 1 a1 t ð8:13Þ This expression may now be evaluated by substituting the data from Section 8.7.4 (1) into Eq. (8.13). Doing this, we get u 5 3000 MW 1 0 MW 3000 MW 300UuðtÞ 1 0 1 30 MW=s Ut or
u 5 2 300UuðtÞ 1 30 MW=s Ut
ð8:14Þ
4. Choosing the “time-step” Δt Let us divide the ramping-up time of 5 seconds by 100 to give a time step of Δt 5 0:05 s
ð8:15Þ
We shall use this time-step to increment the input u and hence the output y.
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The output data
The output data will consist of values of Δy 5 Δ(f 2 fn) that correspond to increments of the input Δu 5 Δ(PG 2 PL) as shown in Diagrams 8.1 and 8.2. The output values for each step are obtained by following the procedure described in Sections 8.6.4 and 8.6.5. Values of f 2 fn can then be plotted against time t. Alternatively, values of f may be plotted against t.
8.7.6
The suggested MATLAB code
The suggested MATLAB code to solve the problem is as follows: % % % %
Solving the “Swing Equation” by the “control diagram” method. Enter the input data in physical units: PNR_P 5 input(‘Please enter the total output of the nonresponsive generation in MW’) PR_P 5 input(‘Please enter the total rated output of the responsive generation In MW’) FN_P 5 input(‘Please enter the system nominal frequency in Hz’) PTRIP_P 5 input(‘Please enter the amount of generation tripped in MW’) TTRIP 5 input(‘Please enter the time of the generation trip in seconds’) H 5 input(‘Please enter the machine inertia constant in seconds’) PLOAD_P 5 input(‘Please enter the total system load in MW’) ALPHA_P 5 input(‘Please enter the “load damping” constant α in per cent per Hz’)
% % %
Now enter the “responsive ramp” data: A0_P 5 input(‘Please enter the value of the “ramp” constant a0in MW’) A1_P 5 input(‘Please enter the value of the “ramp” constant a1in MW per second)
% %
Convert the data as necessary to “per unit” by defining the MVA and f bases:
% MVA_BASE 5 3000/0.8 F_BASE 5 FN_P % %
Now convert all quantities to “per unit” as necessary:
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% PNR 5 PNR_P/MVA_BASE PR 5 PR_P/MVA_BASE FN 5 FN_P/F_BASE PTRIP 5 PTRIP_P/MVA_BASE PLOAD 5 PLOAD_P/MVA_BASE ALPHA 5 ALPHA_P F_BASE/100 A0 5 A0_P/MVA_BASE A1 5 A1_P/MVA_BASE % % %
Now enter the size of the time step in seconds: TSTEP 5 input(‘Please enter the value of the time-step in seconds’)
% % % %
Calculate the values of the parameters K and T of the transfer function using (Eq 8.8a) and (Eq 8.8b): K 5 1/A1 T 5 2 H F0/A1
% % % % % %
In Laplace space Δu(s) 5 Δu/s. We need to multiply this by K/(1 1 s.T) to get Δy(s), of which we can then take the inverse transform to get Δy(t). For the first time increment only: DELTA _OMEGA(1) 5 PTRIP K (EXP(-TSTEP/T) 1) DELTA_F(1) 5 2 PI DELTA_OMEGA(1)
% % % %
Now perform iterations to generate the successive speed changes up to a time limit of 5 seconds: NITER 5 5/TSTEP
% for k 5 2, NITER T 5 k TSTEP DELTA_OMEGA(k) 5 A1 T K (EXP(-TSTEP/T) 1) DELTA_F(k) 5 2 PI DELTA_OMEGA(k) end % %
A graph of the results may now be printed out.
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Instructions for the user
Enter the required input data when prompted by the MATLAB code. The output will be a graph of “system frequency” against time for time t 5 010 seconds.
8.8 Comparison of the “control diagram” approach with some of the other methods available 8.8.1
Introduction
The “control diagram” approach is a sort of hybrid between the “analytical methods” and the “numerical methods”: it contains elements of both. We describe these similarities in turn next.
8.8.2 Similarity of the “control diagram” approach to the “analytical methods” The “control diagram” approach has similarities with the “analytical methods.” There are elements of the “Laplace transform” method because Laplace transforms are needed as part of the “control diagram” approach as well to calculate the change in frequency output as a response to a change in the power balance input.
8.8.3 Similarity of the “control diagram” approach to the “numerical methods” There is also an element of the “numerical methods” in the “control diagram” approach because iterations are required in both methods to find the solution.
8.8.4
Uniqueness of the “control diagram” approach
The “control diagram” approach is unlike the other two groups of methods in its use of the “block diagram” and in the “state-space” approach.
8.9 The next chapter: some important practical applications 8.9.1
Introduction
In the next chapter, we look at some important practical applications. They are as follows: G
G
Calculation of the “rate of change of frequency” (RoCoF) following a loss of generation or demand Calculation of the “system response requirements” following a loss of generation (the “low frequency excursion”)
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Calculation of the “system response requirements” following a loss of demand (the “high frequency excursion”) Calculation of the “system response requirements” during “normal operation” (relatively small variations in the “system frequency”)
These calculations are all important tasks performed regularly every day by control engineers at national control centers around the world. Because of their importance, we give short descriptions of each in preparation for the next chapter as follows:
8.9.2 [I] The calculation of the “rate of change of frequency” following a loss of generation or demand 1. RoCoF relays a. The equipping of generators with “RoCoF relays” Some generators are equipped with “RoCoF relays”. b. Why are “RoCoF” relays installed on generators? “RoCoF” relays may be installed on generators for several reasons: i. Loss of grid/loss of mains/islanding protection One potential use of having “RoCoF” relays on generators is to detect a “loss of grid” condition (also known as “loss of mains” or “islanding”). The purpose of the “RoCoF” relays in this situation is to protect the generator. When the “RoCoF” relay detects a “loss of grid” condition, the generator is “tripped,” that is, disconnected from the main system. ii. The detection of “under-frequency” and “over-frequency” conditions “RoCoF relays” may also be installed on generators to detect both “under-frequency” and “over-frequency” conditions. The purpose of the “RoCoF relay” here is to protect the generator by tripping it before the frequency becomes too low or too high, thus avoiding potentially costly damage to the generator. 2. The control of the RoCoF The “RoCoF” on the power system can be controlled by the scheduling of generation to ensure that there is enough inertia on the system to limit the RoCoF to below generator tripping threshold levels. 3. Calculating the “RoCoF” Calculating the “RoCoF” is achieved straightforwardly by evaluating the right-hand side of the “swing equation,” since “RoCoF” is simply “df/dt.”
8.9.3 [II] The calculation of the “system response requirements” following a loss of generation (a “low frequency excursion”) 1. What is a “low frequency excursion”?
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By “low frequency excursion” we mean that the “system frequency” has deviated, perhaps significantly, from the “normal operating range” and is lower than where it started. 2. Calculating the “system response requirements” following a loss of generation (a “low frequency excursion”) Calculating the “system response requirements” following a loss of generation (a “low frequency excursion”) may be achieved by using the “swing equation” to model sources of generation and demand including “frequency responsive plant” and iterating one of the solution methods in Chapters 68 following a loss of generation to find the amount of “frequency response” needed to maintain the “system frequency” to above the low voltage tolerance required by the “statutory requirements” for that size of generation loss.
8.9.4 [III] The calculation of the “system response requirements” following a loss of demand (a “high frequency excursion”) 1. What is a “high frequency excursion”? By “high frequency excursion” we mean that the “system frequency” has deviated, perhaps significantly, from the “normal operating range” and is higher than where it started. 2. Calculating the “system response requirements” following a loss of demand (a “high frequency excursion”) Calculating the “system response requirements” following a loss of demand (a “high frequency excursion”) may be achieved by using the “swing equation” to model sources of generation and demand including “frequency responsive plant” and iterating one of the solution methods in Chapters 68 following a loss of demand to find the amount of “frequency response” needed to maintain the “system frequency” to below the high voltage tolerance required by the “statutory requirements” for that size of demand loss.
8.9.5 [IV] The calculation of the “system response requirements” during “normal operation” (“relatively small variations” in the “system frequency”) 1. What is “normal operation”? By “normal operation” we mean that the “system frequency” is contained within a narrow band around the “nominal frequency.” There are no “low frequency excursions” or “high frequency excursions.” 2. Calculating the “system response requirements” during “normal operation” (relatively small variations in the “system frequency”) Calculating the “system response requirements” during “normal operation” (relatively small variations in the “system frequency”) may be
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achieved by using the “swing equation” to model sources of generation and demand including “frequency responsive plant” and iterating one of the solution methods in Chapters 68 following a selected small size of loss of generation or demand to find the amount of “frequency response” needed to maintain the “system frequency” both above the low voltage tolerance and below the high voltage tolerance defined by the “normal band of operation” around the “nominal frequency.”
References Books [1] P. Kundur, 22nd reprint Power System Stability and Control. Power System Engineering Series, Electric Power Research Institute, McGraw-Hill Education, 2017. [2] K. Ogata, Modern Control Engineering, fourth ed., Prentice Hall/Pearson Education International, 2002. [3] D.R. Towill, Transfer Function Techniques for Control Engineers, Iliffe Books Ltd, 1970. [4] J.G. Truxal, Control Engineers’ Handbook, McGraw-Hill Book Company, Inc., 1958.